# Basic Algebra/Introduction to Basic Algebra Ideas/Variables and Expressions

Variable
Term
Operation
Expression
Evaluate
Substitute

## Lesson

A variable is a letter or symbol that takes place of a number in Algebra. Common symbols used are ${\displaystyle a}$ , ${\displaystyle x}$ , ${\displaystyle y}$ , ${\displaystyle \theta }$  and ${\displaystyle \lambda }$ . The letters x and y are commonly used, but remember that any other symbols would work just as well.

Variables are used in algebra as placeholders for unknown numbers. If you see "3 + x", don't panic! All this means is that we are adding a number who's value we don't yet know.

Some examples of variables in use:

• ${\displaystyle 3x}$  -- three times of ${\displaystyle x}$ .
• ${\displaystyle 5-y}$  -- five minus ${\displaystyle y}$
• ${\displaystyle 2\div s}$  or ${\displaystyle {\frac {2}{s}}}$ -- 2 divided by ${\displaystyle s}$

A term is a number or a variable or a cluster of numbers and variables multiplied and or divided separated by addition and subtraction.

Examples of terms:

• ${\displaystyle 3+5}$  The terms are 3 and 5.
• ${\displaystyle {\frac {6}{x}}}$  The term is ${\displaystyle 6/x}$ , 6 over ${\displaystyle x}$  is one term, because the operation is division.
• ${\displaystyle 6x+5}$  The terms are 6${\displaystyle x}$  and 5, 6${\displaystyle x}$  and 5 are separate terms because they are separated by a addition or subtraction.

An operation is a thing you do to numbers, like add, subtract, multiply, or divide. You use signs like +, , *, or / for operations.

An expression is two or more terms, with operations between all terms.

Examples of expressions:

• ${\displaystyle 3\div 6}$
• ${\displaystyle 8\times x}$
• ${\displaystyle x\times 6+y}$
• ${\displaystyle a\times b\times c\times d}$

To evaluate an expression, you do the operations to the terms of an expression.

Examples of evaluating expressions:

• ${\displaystyle 3+4}$  evaluates to 7.
• ${\displaystyle 18\div 3}$  evaluates to 6.
• ${\displaystyle 4\times 5-3}$  evaluates to 17.

To evaluate an expression with variables, you substitute (put a thing in the place of an other thing) numbers for the variables.

Examples of substituting: (Substitute 3 for x in these examples.)

• ${\displaystyle x+4}$  is ${\displaystyle 3+4}$ .
• ${\displaystyle 18\div x}$  is ${\displaystyle 18\div 3}$ .
• ${\displaystyle 4\times 5-x}$  is ${\displaystyle 4\times 5-3}$ .

## Example Problems

Evaluate the following expressions

 ${\displaystyle 5\times x}$ When ${\displaystyle x=2}$ ${\displaystyle 5\times 2}$ Substitute 2 for ${\displaystyle x}$ . ${\displaystyle 10}$ Evaluate ${\displaystyle 5\times 2}$  to get the answer.
 ${\displaystyle {\frac {x}{3}}+y}$ When ${\displaystyle x=9}$  and ${\displaystyle y=4}$ ${\displaystyle {\frac {9}{3}}+4}$ Substitute 9 for ${\displaystyle x}$  and substitute 4 for ${\displaystyle y}$ . ${\displaystyle 7}$ Evaluate ${\displaystyle {\frac {9}{3}}+4}$  to get the answer.

## Practice Problems

remember order of operations

Evaluate each expression if ${\displaystyle a}$  = 1, ${\displaystyle b}$  = 2, ${\displaystyle c}$  = 3, and ${\displaystyle d}$  = 5.

1

 ${\displaystyle 5\times b=}$

2

 ${\displaystyle 9\times c=}$

3

 ${\displaystyle c-2=}$

4

 ${\displaystyle d-5=}$

5

 ${\displaystyle {\frac {b}{2}}=}$

6

 ${\displaystyle {\frac {36}{c}}=}$

7

 ${\displaystyle b\times c+2=}$

8

 ${\displaystyle b\times c\times d-5=}$
Evaluate each expression if ${\displaystyle x}$  = 4, ${\displaystyle y}$  = 2, and ${\displaystyle z}$  = 3.

9

 ${\displaystyle x+y=}$

10

 ${\displaystyle 2z=}$

11

 ${\displaystyle xz=}$

12

 ${\displaystyle x+y+z=}$

13

 ${\displaystyle xy+z=}$

14

 ${\displaystyle yz-x=}$

15

 ${\displaystyle {\frac {6}{y}}+z=}$

16

 ${\displaystyle {\frac {2x}{2+y}}=}$
More harder questions: Evaluate each expression if ${\displaystyle x}$  = 5, ${\displaystyle y}$  = 8, and ${\displaystyle z}$  = 9.

17

 ${\displaystyle (2+x)\times y=}$

18

 ${\displaystyle {\frac {3y-9}{5}}=}$

19

 ${\displaystyle {\frac {27}{x+4}}-(y-5)=}$

20

 ${\displaystyle {\frac {z+12}{2x-3}}+y=}$

21

 ${\displaystyle \left({\frac {6x}{2+y}}-z\right)+\left(x-{\frac {z}{3}}\right)=}$

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